Quadrilateral $ABCD$ is a parallelogram. What is the degree measure of angle $A$? [asy] size(100); draw((0,0)--(5,2)--(6,7)--(1,5)--cycle); draw((5,2)--(7.5,3)); draw(Arc((5,2),1,20,80)); label("$D$",(0,0),SW); label("$C$",(5,2),SE); label("$B$",(6,7),NE); label("$A$",(1,5),NW); label("$60^\circ$",(6.3,2.8), N); [/asy]
Answer: All angle measures will be in degrees. $\angle DCB = 180 - 60 = 120$, and since opposite angles in a parallelogram are equal, we have $\angle A = \angle DCB = 120$. Hence the degree measure of $\angle A$ is $\boxed{120}$.